Analysis of the Navier-Stokes regularity problem

众所周知，Navier-Stokes方程的“规律性”问题是我们这个时代最困难、最有趣、最重要的数学问题之一。它是克莱数学研究所在2000年提出的七个著名的千禧年奖问题之一，其中只有一个（庞加莱猜想）已经得到解决，并且一直是英国和国外科学机构（如ERC（欧盟）和NSF（美国））颁发的众多国际会议和赠款的焦点。然而，自1934年以来，纳维-斯托克斯方程的相关问题一直吸引着数学家的注意，当时法国数学家让·勒雷（Jean Leray）取得了迄今为止最重要的进展。早在1822年就导出的非线性偏微分方程（PDEs）的Navier-Stokes系统，被认为给出了近似决定空气和水等流体运动的规则。这有很多问题，不仅是方程是否准确地描述了潜在的物理，更重要的是，方程本身是否能给出物理上相关的预测。它们更可能预测潜在的古怪和纯粹的数学运动，其中包含被称为数学“奇点”的不稳定行为。当然，根据我们对自然的经验，我们预计，如果没有显著的外部强迫，这种情况在现实中永远不会发生，因此，知道奇点是否可以在数学上形成，将对模型的有效性产生根本性的影响。勒雷证明了这些方程总是提供至少一种可预测的运动，从那时起，数学家们一直在试图确定这些方程是否实际上包含数学奇点。此外，这些是分析非线性偏微分方程（往往基于物理学、生物学、金融学和社会学等领域发生的现象）时的相当典型的问题，对这个问题的解决也将对开发强大的分析工具产生重大的数学影响。事实上，EPSRC最近在英国牛津和爱丁堡建立了两个大型的资助研究中心，用于分析和pde，以及两个博士培训中心，试图提高英国在这些数学领域的竞争力。此外，在英国，尤其是在牛津、沃里克、伦敦和苏塞克斯，已经做了很多工作来解决和提高人们对纳维-斯托克斯和其他解析非线性偏微分方程问题的兴趣。该项目旨在简化最近解决Navier-Stokes正则性问题的大量工作，并确定有效的未来方向，同时将问题本身和数学奇点的概念置于更令人满意的环境中。这些目标大致可以分为三个部分：(1)描述潜在的奇点；(2)理解当前技术的障碍；(3)在推导方程时使用的建模假设的背景下探索“奇点”。这些努力将对潜在奇点的性质给出更全面和定性的描述。然后，人们可以利用这种理解来集中精力构建奇点的明确示例或排除奇点形成的可能性，以及将数学问题本身置于令人满意的建模环境中。

The question of "regularity" for the Navier-Stokes equations is well-known to be one of the most difficult, interesting and important mathematical questions of our time. It is the subject of one of the seven famous Millennium Prize Problems posed by the Clay Mathematics Institute in 2000, of which only one (the Poincare Conjecture) has since been resolved, and has been as well the focus of numerous international conferences and grants awarded in the UK and abroad by scientific agencies such as the ERC (EU) and NSF (USA). However the relevant questions for the Navier-Stokes equations have absorbed the attention of mathematicians since 1934 when the key significant advance to date was made by French mathematician Jean Leray. The Navier-Stokes system of nonlinear partial differential equations (PDEs), derived as early as 1822, is thought to give rules that approximately determine the motion of fluids such as air and water. There are many issues with this, not only as to whether the equations accurately describe the underlying physics but perhaps more fundamentally whether the equations themselves could give physically relevant predictions. They may rather predict potentially eccentric and purely mathematical motions containing erratic behaviors known as mathematical "singularities". Of course, from our experience with nature we expect that without significant external forcing this would never happen in reality, and so knowing whether or not singularities could form mathematically would have fundamental implications for the efficacy of the model. Leray proved that the equations always provide at least one predicted motion, and mathematicians have been trying to determine whether these could in fact contain mathematical singularities ever since. These are moreover quite typical questions in the analysis of nonlinear PDEs (which tend to be based on phenomena occurring in areas such as physics, biology, finance and sociology) and a resolution to this question would also have a significant mathematical impact with regard to development of robust analytical tools. The EPSRC has in fact recently developed two large funded research centers for analysis and PDEs in the UK, in Oxford and Edinburgh, as well as two doctoral training centers, to try to boost the UK's competitiveness in these mathematical areas. Moreover, much work has been done in the UK, most notably in Oxford, Warwick, London and Sussex, to address and boost interest in the Navier-Stokes and other analytical nonlinear PDE problems.This project aims specifically to streamline the copious recent efforts to resolve the Navier-Stokes regularity problem and determine effective future directions, as well as to put the question itself, and the idea of mathematical singularities, into a more satisfying context. These goals can be grouped roughly into three parts: (1) characterizing potential singularities, (2) understanding obstacles to current techniques and (3) exploring "singularity" in the context of the modeling assumptions used in deriving the equations. These efforts will give a more comprehensive and qualitative description of the nature of potential singularities. One can then use this understanding to focus efforts to either construct explicit examples of singularities or rule out the possibility of their formation, as well as to put the mathematical question itself into a satisfying modeling context.